Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 520219 9 pageshttpdxdoiorg1011552013520219

Research ArticleOptimal Rate of Convergence fora Nonstandard Finite Difference Galerkin MethodApplied to Wave Equation Problems

Pius W M Chin

Department of Mathematics and Applied Mathematics University of Pretoria Pretoria 0002 South Africa

Correspondence should be addressed to Pius W M Chin piuschinupacza

Received 6 August 2013 Revised 11 November 2013 Accepted 14 November 2013

Academic Editor Song Cen

Copyright copy 2013 Pius W M Chin This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The optimal rate of convergence of the wave equation in both the energy and the 1198712-norms using continuous Galerkin method iswell known We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite differencemethod in the time and the continuous Galerkin method in the space variables We show that for sufficiently smooth solutionthe maximal error in the 1198712-norm possesses the optimal rate of convergence 119874(ℎ2 + (Δ119905)2) where h is the mesh size and Δ119905 is thetime step size Furthermore we show that this scheme replicates the properties of the exact solution of the wave equation Somenumerical experiments should be performed to support our theoretical analysis

1 Introduction

Most physical phenomena such as the acoustics electromag-netic and elastic problems aremodeled by the wave equationThequalitative solution of themodel in the spacetime domainis always a very delicate but a fundamental issue that needscareful study Our point of departure of this paper is toconsider the following model of the wave equation find119906(119909 119905) such that

1205972119906

1205971199052minus Δ119906 = 119891 in (119909 119905) isin Ω times (0 119879) (1)

119906 (119909 0) = 1199060(119909) (2)

120597119906 (119909 0)

120597119905= 1199061(119909) 119909 isin Ω (3)

where Ω is a smooth bounded domain in R2 with smoothboundary 120597Ω Problem (1)ndash(3) consists of constant coef-ficients and 119891(119909 119905) the source term 119906

0(119909) and 119906

1(119909) are

prescribed as the initial data Furthermore (0 119879) is taken tobe a finite time interval and the boundary conditions satisfiedby 119906(119909 119905) are given by

119906 (119909 119905) = 0 (119909 119905) isin 120597Ω times (0 119879) (4)

The methods which have been heavily used for the study ofthe wave equation (1)ndash(4) in physical life are the continuousas well as the discontinuous Galerkin methods see [1 2] formore detailsThe reason for these methods may be due to theway they dealwith heterogeneousmedia and arbitrary shapedgeometric objects represented by unstructured grids

The advantages of the continuous Galerkin method areenormous Firstly the convergence theory of this methodis based on lower regularity (differentiability) requirementsthan the finite difference and the spectral methods Secondlythe method retains the important energy conservation prop-erties provided by the discrete version of the initialboundaryvalued problem such as the one under considerationThirdlythe computation and the analysis from the Galerkin methodcould be extended in the approximation of the nonlinearwave equation Furthermore the method could be applicableto problems of any desired order of accuracy Formore detailson these advantages see in [3ndash5]

The apriori error estimate for continuous Galerkinapproximation of the wave equation (1)ndash(4) was first derivedby Dupont [6] and later improved by Baker [7] both for con-tinuous and discrete time schemes Gekeler [8] analyzed gen-eral multistep methods for the time discretization of second-order hyperbolic equation when a Galerkin procedure is

2 Journal of Applied Mathematics

used in spaceThe nonclassical finite element treatment of thewave equation can be seen in Johnson [9] and Richter [10]

In this paper we exploit and present a reliable techniqueconsisting of coupling the nonstandard finite difference(NSFD) method in time and the continuous Galerkin (CG)method in the space variables A similar approach was donefor the first time using parabolic problems more specificallythe diffusion equations in the nonsmooth domain as seen in[11] The NSFD method was initiated by Mickens in [12] andmajor contributions to the foundation of the NSFD methodcould be seen in [13 14] Since its initiation theNSFDmethodhas been extensively applied to many concrete problems inengineering and science see [12 15 16] for an overview Thispaper compliments the technique used in [11] The techniqueis geared toward obtaining a sufficiently smooth solution themaximal error in the 1198712-norm and to show that the erroracross the entire interval convergences optimally as 119874(ℎ2 +Δ1199052) where ℎ is the mesh size and Δ119905 is the time step size

The reliability of this technique comes from the fact that theNSFD-CGmethod preserves both the energy features and thehyperbolicity of the exact solution of the wave equation (1)ndash(4)

The organization of the paper is as follows Under Sec-tion 2 we review some of the useful spaces and their notationsneeded in the paper In Section 3 we gather essential toolsnecessary to prove the main result of the paper We presentin Section 4 a reliable scheme NSFD-CG and show thatthe numerical solution obtained from this scheme attainsthe optimal convergence rate in the energy as well as inthe 1198712-norms Furthermore we show that the scheme underconsideration replicates the properties of the exact solutionSection 5 is devoted to some numerical experiments usinga numerical example which confirms the optimal rate ofconvergence of the solution proved analytically in Section 4The concluding remarks are given in Section 6 and theseunderline how the work fits in the existing literature and alsohow it can be extended for further work

2 Notations

In this section we will review some of the spaces which wewill be using in the paper together with their notations andpossibly properties For 119904 ge 0119867119904(Ω) will denote the Sobolevspace of real-valued functions onΩ and the norm on119867119904(Ω)will be denoted by sdot

119904 See [17] for the definitions and the

relevant properties of these spaces In a particular case where119904 = 0 the space1198670(Ω) = 119871

2(Ω) and its inner product together

with the norm will be denoted respectively by

(119906 V) = intΩ

119906V 119889119909 119906 V isin 1198712(Ω)

1199061198712(Ω)

= (119906 119906)12 119906 isin 119871

2(Ω)

(5)

In addition 119862infin0(Ω) will denote the space of infinitely dif-

ferentiable functions with support compactly contained inΩ The space 119867

1

0(Ω) will denote the subspace of 1198671(Ω)

obtained by completing 119862infin

0(Ω) with respect to the norm

sdot 1 Following [17] for 119883 a Hilbert space we will more

generally use the Sobolev space 119867119904[(0 119879) 119883] where 119904 ge 0

and in the case where 119904 = 0 we will have 1198670[(0 119879) 119883] equiv

1198712[(0 119879) 119883] with norm

V1198712[(0119879)119883] = (int

119879

0

V (sdot 119905)119883119889119905)

12

(6)

In practice 119883 will be the Sobolev space 119867119898(Ω) or 1198671198980(Ω)

Associated with (1) is the bilinear form

119886 (119906 V) = intΩ

nabla119906nablaV 119889119909 119906 V isin 1198671(Ω) (7)

119886(sdot sdot) will be symmetric and positive definite that is

119886 (119906 V) = 119886 (V 119906) 119886 (119906 119906) ge 0 (8)

3 The Continuous Galerkin Method

Having the previously mentioned notations in place weproceed under this section to gather essential tools necessaryto prove the main result of our paper We begin by stating thefollowing weak problem of (1)ndash(4)

Find 119906 isin 1198712[(0 119879)119867

1

0(Ω)] given 119891 isin 119871

2[(0 119879) 119871

2(Ω)]

such that

(1205972119906 (sdot 119905)

1205971199052 V) minus 119886 (119906 (sdot 119905) V) = (119891 (sdot 119905) V)

forallV isin 1198671

0(Ω) 119905 ge 0

(119906 (sdot 0) V) = (1199060 V)

(120597119906 (sdot 0)

120597119905 V) = (119906

1 V)

(9)

See [8] for the existence and the uniqueness of a solution119906 of (9) Hence forth in appropriate places to followadditional conditions on the regularity of 119906 which guaranteethe convergence results will be imposed We continue nextby providing the framework for stating the discrete version of(9) To this end we letT

ℎbe a regular family of triangulations

of Ω consisting of compatible triangles 119879 of diameter ℎ119879le

ℎ see [17] for more For each mesh size Tℎ we associate

the finite element space 119881ℎof continuous piecewise linear

function that are zero on the boundary

119881ℎ= Vℎisin 1198620(Ω) V

ℎ|120597Ω

= 0 Vℎ|119879isin 1198751 forall119879 isin T

ℎ (10)

where 1198751is the space of polynomials of degree less than or

equal to 1 and 119881ℎis a finite dimensional subspace of the

Sobolev space 11986710(Ω) It is well known that 119881

ℎparametrized

by ℎ isin (0 1) possesses the following approximation proper-ties there exists a constant 119862 such that if V isin 1198671

0(Ω) cap119867

2(Ω)

we have

inf120594isin119881ℎ

1003817100381710038171003817V minus 120594

1003817100381710038171003817 + ℎ1003817100381710038171003817V minus 120594

10038171003817100381710038171 le 119862ℎ

2V2 (11)

Journal of Applied Mathematics 3

By the use of the energymethod andGronwallrsquos Lemma thereexists a discrete Galerkin solution 119906

ℎisin 119881ℎsuch that

(1205972119906ℎ

1205971199052 Vℎ) minus 119886 (119906

ℎ Vℎ) = (119891 V

ℎ) forallV

ℎisin 119881ℎ 119905 isin [0 119879]

(119906ℎ Vℎ) = (119875

ℎ1199060 Vℎ)

(120597119906ℎ

120597119905 Vℎ) = (119875

ℎ1199061 Vℎ)

(12)

where 119875ℎdenote the 1198712-projection onto 119881

ℎ Furthermore we

let 119906 isin 1198672(Ω) and we define the Galerkin projection 120587

ℎ119906 of

119906 in 119881ℎby requiring that

119886 (120587ℎ119906 V) = 119886 (119906 V) forallV isin 119881

ℎ (13)

120597119896(120587ℎ119906)

120597119905119896= 120587ℎ(120597119896119906

120597119905119896) 119896 = 0 1 2 119905 isin [0 119879] (14)

The previous essential tools lead to the immediate conse-quence of the approximation properties (11)

Lemma 1 Let 119906 be the solution of (9) Then there existsa unique mapping 120587

ℎ119906 isin 119871

2[(0 119879) 119881

ℎ] which satisfies

(13) Furthermore if for some integer 119896 ge 0 120597119896119906120597119905119896 isin

119871119901[(0 119879)119867

2(Ω)] then

120597119896(120587ℎ119906)

120597119905119896isin 119871119901[(0 119879) 119881

ℎ]

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

[119906 minus 120587ℎ119906]

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198712(Ω)]

le 119862ℎ2

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

119906

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198672(Ω)]

(15)

for some constant 119862 independent of the mesh size

4 Coupled Nonstandard Finite Difference andContinuous Galerkin Methods

Instead of the continuous Galerkin method summarizedpreviously we present in this section a reliable schemeNSFD-CG consisting of the nonstandard finite difference methodin time and the continuous Galerkin method in the spacevariable We show that the numerical solution obtained fromthe scheme NSFD-CG attained the optimal convergence inboth the energy and 119871

2-norms We proceed in this regardby letting the step size 119905

119899= 119899Δ119905 for 119899 = 0 1 2 119873 For a

sufficiently smooth function V(119909 119905) we set

(120597

120597119905)

119896

V119899= (

120597

120597119905)

119896

V (sdot 119905119899) V

119899= V (sdot 119905

119899) 119896 ge 0 (16)

With this we proceed to find the NSFD-CG approximation119880119899

ℎ such that 119880119899

ℎasymp 119906119899

ℎat discrete time 119905

119899 That is find a

sequence 119880119899ℎ119873

119899=0in 119881ℎsuch that

(1205752119880119899

ℎ Vℎ) + 119886 (119880

119899

ℎ Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(17)

where

1205752119880119899

ℎ=119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

(120595 (Δ119905))2

119899 = 1 2 119873 minus 1 (18)

and 120595(Δ119905) = 2 sin(Δ1199052) restricted between 0 lt 120595(Δ119905) lt 1The initial conditions 1198800

ℎisin 119881ℎand 1198801

ℎisin 119881ℎare given by

(1198800

ℎ Vℎ) = (119875

ℎ1199060 Vℎ) (19)

(1198801

ℎ Vℎ) = (119880

0

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ119905))

2

20

ℎ Vℎ)

forallVℎisin 119881ℎ

(20)

where 0ℎisin 119881ℎis defined by

(0

ℎ Vℎ) = (119891

0 Vℎ) minus 119886 (119906

0 Vℎ) forallV

ℎisin 119881ℎ (21)

If 119891 = 0 in (1) we will have in view of (17) an exact scheme

(119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

4sin2 (Δ1199052) Vℎ) + (nabla119880

119899

ℎ nablaVℎ) = 0 (22)

which according to Mickens [12] replicates both the energypreserving features and the properties of the exact solution(1)ndash(4) In order to state and prove the main result we needa framework on which this result is based To this end weproceed by decomposing the error denoted by 119890119899 = 119906

119899minus 119880119899

ℎ

into the following error equation

119890119899= 119906119899minus 119908119899

ℎ+ 119908119899

ℎminus 119880119899

ℎ= 120578119899minus 120601119899 (23)

where 119908119899ℎ= 120587ℎ119906119899isin 119881ℎis the Galerkin projector of 119906119899

Due to the regularity assumptionmentioned earlier the exactsolution 119906 of (1)ndash(4) satisfies

(1205972119906119899

1205971199052 Vℎ) + 119886 (119906

119899 Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873

(24)

Subtracting (17) from (24) and using some properties of theGalerkin projection in the space we have

(1205752119908119899

ℎminus 1205752119880119899

ℎ Vℎ) + 119886 (119908

119899

ℎminus 119906119899 Vℎ)

= (1205752119908119899

ℎminus1205972119906119899

1205971199052 Vℎ)

(25)

from where we obtain in view of (23)

(1205752120601119899 Vℎ) + (120601

119899 Vℎ) = (119903

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(26)

where 119903119899 = 1205752119908119899

ℎminus12059721199061198991205971199052 In view of the necessity for error

bound we set

119903119899=

1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1

1206011minus 1206010

(120595 (Δ119905))2

for 119899 = 0

(27)

4 Journal of Applied Mathematics

from where we define

119877119899= 120595 (Δ119905)

119899

sum

119898=0

119903119898 (28)

The previously mentioned framework leads to the followingmain result

Theorem 2 Let 119906 be the solution of (9) and 119880119899

ℎ119873

119899=0in

119881ℎ

a sequence defined by (17)ndash(21) Suppose that 119906 isin

1198712[(0 119879)119867

2(Ω)] 120597119906120597119905 isin 119871

2[(0 119879)119867

2(Ω)] and 120597119896119906120597119905119896 isin

1198712[(0 119879) 119871

2(Ω)] for 119896 = 3 4 Then there exists a constant

119862 gt 0 independent of Δ119905 and the mesh refinement size ℎ

max0le119899le119873

1003817100381710038171003817119906 (sdot Δ119905) minus 119880119899

ℎ

10038171003817100381710038170le 119862 [ℎ

2+ (Δ119905)

2] (29)

Furthermore the discrete solution replicates all the propertiesof solution of the hyperbolic equation in the limiting case of thespace independent equation

We prove the previousTheorem 2 thanks to the followingseries of results

Proposition 3 The following result holds for a constant 119862 gt 0

that is independent of Δ119905 and the mesh refinement size ℎ

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862(100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+120595 (Δ119905) max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(30)

Proof In view of (23) we have the following relation usingtriangular inequality

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

(31)

We bound 120601119899 in (31) by first taking partial sums of the firstterm of (26) seconded by adding the remaining terms from119899 = 1 to119898 for 1 le 119898 le 119873 minus 1 and multiplying both sides by120595(Δ119905) yields

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) minus (

1206011minus 1206010

120595 (Δ119905) Vℎ) + 120595 (Δ119905)

119898

sum

119899=1

119886 (120601119898 Vℎ)

= 120595 (Δ119905)

119898

sum

119899=1

(119903119899 Vℎ)

(32)

In view of (28) we can define

Φ119898= 120595 (Δ119905)

119898

sum

119899=1

120601119898

Φ0= 0

(33)

and using this in (32) we have

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) + 119886 (Φ

119898 Vℎ) = (119877

119898 Vℎ) forallV

ℎisin 119881ℎ

(34)

If we choose Vℎ= 120601119898+1

+ 120601119898isin 119881ℎin (34) and multiply the

result by 120595(Δ119905) this yields10038171003817100381710038171003817120601119898+110038171003817100381710038171003817

2

0minus10038171003817100381710038171206011198981003817100381710038171003817

2

0+ 120595 (Δ119905) 119886 (Φ

119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=1

(119877119898 120601119898+1

+ 120601119898)

for 0 le 119898 le 119873 minus 1

(35)

Summing this from 119898 = 0 to 119898 = 119899 minus 1 for 1 le 119899 le 119873 minus 1

gives

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+1

+ 120601119898)

(36)

Since 119886 is symmetricΦ0 = 0 and

Φ119898+1

minus Φ119898minus1

= 120595 (Δ119905) (120601119898+ 120601119898+1

) 119898 = 1 2 119873 minus 1

(37)

then we deduce in view of the third term of (36)

120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

+ Φ119898minus1

)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

)

minus

119899minus1

sum

119898=0

119886 (Φ119898+1

Φ119898minus1

)

= 119886 (Φ119899minus1

Φ119899)

(38)

By symmetry and coercivity properties of 119886 and the fact that

Φ119899minus Φ119899minus1

= 120595 (Δ119905) 120601119899 for 119899 = 1 2 119873 (39)

we have in view of (38)

(120595 (Δ119905))2

2119886 (120601119899 120601119899)

=1

2119886 (Φ119899minus Φ119899minus119894 Φ119899minus Φ119899minus1

)

=1

2119886 (Φ119899 Φ119899) minus 119886 (Φ

119899 Φ119899minus1

)

+1

2119886 (Φ119899minus1

Φ119899minus1

)

(40)

Journal of Applied Mathematics 5

and so

(120595 (Δ119905))2

2119886 (120601119899 120601119899) ge minus119886 (Φ

119899 Φ119899minus1

) (41)

and this together with (36) yields

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2119886 (120601119899 120601119899)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

for 1 le 119899 le 119873

(42)

By Poincare inequality together with the continuity andcoercivity of 119886 we have the following inequality

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2

10038171003817100381710038171206011198991003817100381710038171003817

2

0

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

(43)

We suppose at this stage that ℎ and Δ119905 satisfy the CFLcondition such that 120595(Δ119905)ℎ lt 1 With this condition theprevious inequality becomes

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

1 le 119899 le 119873

(44)

from where we have 12 lt 119862Δ119905= 1 minus (120595(Δ119905))

22ℎ2lt 1 and

12 lt 119862Δ119905lt 1 By the use of Cauchy-Schwarz inequality on

(44) and some algebraic manipulations we have

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+ 120601119898+1

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

(100381710038171003817100381712060111989810038171003817100381710038170

+10038171003817100381710038171003817120601119898+1100381710038171003817100381710038170

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 2120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

max119898=0119899

100381710038171003817100381712060111989810038171003817100381710038170

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+119862Δ119905

2max119898=1119899

10038171003817100381710038171206011198981003817100381710038171003817

2

0

+2

119862Δ119905

(120595(Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

for 0 le 119898 le 119899

(45)

and since the right-hand side is independent of 119899 then

(119862Δ119905minus119862Δ119905

2)10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(46)

which then implies that

119862Δ119905

2

10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(47)

Furthermore we have

10038171003817100381710038171206011198991003817100381710038171003817

2

0le

2

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0+4(120595 (Δ119905))

2

119862Δ119905

(

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(48)

and hence the following result

max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

le radic2

119862Δ119905

100381710038171003817100381710038171206010100381710038171003817100381710038170

+2120595 (Δ119905)

119862Δ119905

119873minus1

sum

119899=1

100381710038171003817100381711987711989910038171003817100381710038170

(49)

With (49) and the fact that100381710038171003817100381710038171206010100381710038171003817100381710038170

le100381710038171003817100381710038171198900100381710038171003817100381710038170

+100381710038171003817100381710038171205780100381710038171003817100381710038170

(50)

we have in viewof (31) the desired estimate of the proposition

We now proceed to bound the term containing 119877119899 on

the right-hand side of Proposition 3 We achieve this byestimating in the 1198712-norm the function 119903119899 for the cases 119899 = 0

and follow by the case when 119899 ge 1

Lemma 4 There holds100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862[(120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

]

(51)

with a constant 119862 gt 0 that is independent of ℎ and the meshsize

Proof In view of (27) we have 1199030 = (120595(Δ119905))minus2(1206011minus 1206010) We

estimate 1206011 minus 12060100by taking V

ℎisin 119881ℎarbitrary as follows

(1206011minus 1206010 Vℎ)

= (1199081

ℎminus 1198801

ℎ Vℎ) minus (119908

0

ℎminus 1198800

ℎ Vℎ)

= ((120587ℎminus 119868) (119906

1minus 1199060) Vℎ) + (119906

1minus 1198801 Vℎ)

(52)

where we have used (1199060minus1198800

ℎ Vℎ) = (119906

0minus119875ℎ1199060 V) = 0 in view

of (19) It now follows from (52) that1003816100381610038161003816((120587ℎ minus 119868) (1199061 minus 1199060) Vℎ)

1003816100381610038161003816

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

(120597

120597119905(120587ℎminus 119868) 119906 (sdot 119904) V

ℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By Lemma 1

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

((120587ℎminus 119868)

120597119906 (sdot 119904)

120597119905 Vℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By (18)

le 119862120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

1003817100381710038171003817Vℎ10038171003817100381710038170

(53)

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

used in spaceThe nonclassical finite element treatment of thewave equation can be seen in Johnson [9] and Richter [10]

In this paper we exploit and present a reliable techniqueconsisting of coupling the nonstandard finite difference(NSFD) method in time and the continuous Galerkin (CG)method in the space variables A similar approach was donefor the first time using parabolic problems more specificallythe diffusion equations in the nonsmooth domain as seen in[11] The NSFD method was initiated by Mickens in [12] andmajor contributions to the foundation of the NSFD methodcould be seen in [13 14] Since its initiation theNSFDmethodhas been extensively applied to many concrete problems inengineering and science see [12 15 16] for an overview Thispaper compliments the technique used in [11] The techniqueis geared toward obtaining a sufficiently smooth solution themaximal error in the 1198712-norm and to show that the erroracross the entire interval convergences optimally as 119874(ℎ2 +Δ1199052) where ℎ is the mesh size and Δ119905 is the time step size

The reliability of this technique comes from the fact that theNSFD-CGmethod preserves both the energy features and thehyperbolicity of the exact solution of the wave equation (1)ndash(4)

The organization of the paper is as follows Under Sec-tion 2 we review some of the useful spaces and their notationsneeded in the paper In Section 3 we gather essential toolsnecessary to prove the main result of the paper We presentin Section 4 a reliable scheme NSFD-CG and show thatthe numerical solution obtained from this scheme attainsthe optimal convergence rate in the energy as well as inthe 1198712-norms Furthermore we show that the scheme underconsideration replicates the properties of the exact solutionSection 5 is devoted to some numerical experiments usinga numerical example which confirms the optimal rate ofconvergence of the solution proved analytically in Section 4The concluding remarks are given in Section 6 and theseunderline how the work fits in the existing literature and alsohow it can be extended for further work

2 Notations

In this section we will review some of the spaces which wewill be using in the paper together with their notations andpossibly properties For 119904 ge 0119867119904(Ω) will denote the Sobolevspace of real-valued functions onΩ and the norm on119867119904(Ω)will be denoted by sdot

119904 See [17] for the definitions and the

relevant properties of these spaces In a particular case where119904 = 0 the space1198670(Ω) = 119871

2(Ω) and its inner product together

with the norm will be denoted respectively by

(119906 V) = intΩ

119906V 119889119909 119906 V isin 1198712(Ω)

1199061198712(Ω)

= (119906 119906)12 119906 isin 119871

2(Ω)

(5)

In addition 119862infin0(Ω) will denote the space of infinitely dif-

ferentiable functions with support compactly contained inΩ The space 119867

1

0(Ω) will denote the subspace of 1198671(Ω)

obtained by completing 119862infin

0(Ω) with respect to the norm

sdot 1 Following [17] for 119883 a Hilbert space we will more

generally use the Sobolev space 119867119904[(0 119879) 119883] where 119904 ge 0

and in the case where 119904 = 0 we will have 1198670[(0 119879) 119883] equiv

1198712[(0 119879) 119883] with norm

V1198712[(0119879)119883] = (int

119879

0

V (sdot 119905)119883119889119905)

12

(6)

In practice 119883 will be the Sobolev space 119867119898(Ω) or 1198671198980(Ω)

Associated with (1) is the bilinear form

119886 (119906 V) = intΩ

nabla119906nablaV 119889119909 119906 V isin 1198671(Ω) (7)

119886(sdot sdot) will be symmetric and positive definite that is

119886 (119906 V) = 119886 (V 119906) 119886 (119906 119906) ge 0 (8)

3 The Continuous Galerkin Method

Having the previously mentioned notations in place weproceed under this section to gather essential tools necessaryto prove the main result of our paper We begin by stating thefollowing weak problem of (1)ndash(4)

Find 119906 isin 1198712[(0 119879)119867

1

0(Ω)] given 119891 isin 119871

2[(0 119879) 119871

2(Ω)]

such that

(1205972119906 (sdot 119905)

1205971199052 V) minus 119886 (119906 (sdot 119905) V) = (119891 (sdot 119905) V)

forallV isin 1198671

0(Ω) 119905 ge 0

(119906 (sdot 0) V) = (1199060 V)

(120597119906 (sdot 0)

120597119905 V) = (119906

1 V)

(9)

See [8] for the existence and the uniqueness of a solution119906 of (9) Hence forth in appropriate places to followadditional conditions on the regularity of 119906 which guaranteethe convergence results will be imposed We continue nextby providing the framework for stating the discrete version of(9) To this end we letT

ℎbe a regular family of triangulations

of Ω consisting of compatible triangles 119879 of diameter ℎ119879le

ℎ see [17] for more For each mesh size Tℎ we associate

the finite element space 119881ℎof continuous piecewise linear

function that are zero on the boundary

119881ℎ= Vℎisin 1198620(Ω) V

ℎ|120597Ω

= 0 Vℎ|119879isin 1198751 forall119879 isin T

ℎ (10)

where 1198751is the space of polynomials of degree less than or

equal to 1 and 119881ℎis a finite dimensional subspace of the

Sobolev space 11986710(Ω) It is well known that 119881

ℎparametrized

by ℎ isin (0 1) possesses the following approximation proper-ties there exists a constant 119862 such that if V isin 1198671

0(Ω) cap119867

2(Ω)

we have

inf120594isin119881ℎ

1003817100381710038171003817V minus 120594

1003817100381710038171003817 + ℎ1003817100381710038171003817V minus 120594

10038171003817100381710038171 le 119862ℎ

2V2 (11)

Journal of Applied Mathematics 3

By the use of the energymethod andGronwallrsquos Lemma thereexists a discrete Galerkin solution 119906

ℎisin 119881ℎsuch that

(1205972119906ℎ

1205971199052 Vℎ) minus 119886 (119906

ℎ Vℎ) = (119891 V

ℎ) forallV

ℎisin 119881ℎ 119905 isin [0 119879]

(119906ℎ Vℎ) = (119875

ℎ1199060 Vℎ)

(120597119906ℎ

120597119905 Vℎ) = (119875

ℎ1199061 Vℎ)

(12)

where 119875ℎdenote the 1198712-projection onto 119881

ℎ Furthermore we

let 119906 isin 1198672(Ω) and we define the Galerkin projection 120587

ℎ119906 of

119906 in 119881ℎby requiring that

119886 (120587ℎ119906 V) = 119886 (119906 V) forallV isin 119881

ℎ (13)

120597119896(120587ℎ119906)

120597119905119896= 120587ℎ(120597119896119906

120597119905119896) 119896 = 0 1 2 119905 isin [0 119879] (14)

The previous essential tools lead to the immediate conse-quence of the approximation properties (11)

Lemma 1 Let 119906 be the solution of (9) Then there existsa unique mapping 120587

ℎ119906 isin 119871

2[(0 119879) 119881

ℎ] which satisfies

(13) Furthermore if for some integer 119896 ge 0 120597119896119906120597119905119896 isin

119871119901[(0 119879)119867

2(Ω)] then

120597119896(120587ℎ119906)

120597119905119896isin 119871119901[(0 119879) 119881

ℎ]

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

[119906 minus 120587ℎ119906]

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198712(Ω)]

le 119862ℎ2

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

119906

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198672(Ω)]

(15)

for some constant 119862 independent of the mesh size

4 Coupled Nonstandard Finite Difference andContinuous Galerkin Methods

Instead of the continuous Galerkin method summarizedpreviously we present in this section a reliable schemeNSFD-CG consisting of the nonstandard finite difference methodin time and the continuous Galerkin method in the spacevariable We show that the numerical solution obtained fromthe scheme NSFD-CG attained the optimal convergence inboth the energy and 119871

2-norms We proceed in this regardby letting the step size 119905

119899= 119899Δ119905 for 119899 = 0 1 2 119873 For a

sufficiently smooth function V(119909 119905) we set

(120597

120597119905)

119896

V119899= (

120597

120597119905)

119896

V (sdot 119905119899) V

119899= V (sdot 119905

119899) 119896 ge 0 (16)

With this we proceed to find the NSFD-CG approximation119880119899

ℎ such that 119880119899

ℎasymp 119906119899

ℎat discrete time 119905

119899 That is find a

sequence 119880119899ℎ119873

119899=0in 119881ℎsuch that

(1205752119880119899

ℎ Vℎ) + 119886 (119880

119899

ℎ Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(17)

where

1205752119880119899

ℎ=119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

(120595 (Δ119905))2

119899 = 1 2 119873 minus 1 (18)

and 120595(Δ119905) = 2 sin(Δ1199052) restricted between 0 lt 120595(Δ119905) lt 1The initial conditions 1198800

ℎisin 119881ℎand 1198801

ℎisin 119881ℎare given by

(1198800

ℎ Vℎ) = (119875

ℎ1199060 Vℎ) (19)

(1198801

ℎ Vℎ) = (119880

0

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ119905))

2

20

ℎ Vℎ)

forallVℎisin 119881ℎ

(20)

where 0ℎisin 119881ℎis defined by

(0

ℎ Vℎ) = (119891

0 Vℎ) minus 119886 (119906

0 Vℎ) forallV

ℎisin 119881ℎ (21)

If 119891 = 0 in (1) we will have in view of (17) an exact scheme

(119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

4sin2 (Δ1199052) Vℎ) + (nabla119880

119899

ℎ nablaVℎ) = 0 (22)

which according to Mickens [12] replicates both the energypreserving features and the properties of the exact solution(1)ndash(4) In order to state and prove the main result we needa framework on which this result is based To this end weproceed by decomposing the error denoted by 119890119899 = 119906

119899minus 119880119899

ℎ

into the following error equation

119890119899= 119906119899minus 119908119899

ℎ+ 119908119899

ℎminus 119880119899

ℎ= 120578119899minus 120601119899 (23)

where 119908119899ℎ= 120587ℎ119906119899isin 119881ℎis the Galerkin projector of 119906119899

Due to the regularity assumptionmentioned earlier the exactsolution 119906 of (1)ndash(4) satisfies

(1205972119906119899

1205971199052 Vℎ) + 119886 (119906

119899 Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873

(24)

Subtracting (17) from (24) and using some properties of theGalerkin projection in the space we have

(1205752119908119899

ℎminus 1205752119880119899

ℎ Vℎ) + 119886 (119908

119899

ℎminus 119906119899 Vℎ)

= (1205752119908119899

ℎminus1205972119906119899

1205971199052 Vℎ)

(25)

from where we obtain in view of (23)

(1205752120601119899 Vℎ) + (120601

119899 Vℎ) = (119903

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(26)

where 119903119899 = 1205752119908119899

ℎminus12059721199061198991205971199052 In view of the necessity for error

bound we set

119903119899=

1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1

1206011minus 1206010

(120595 (Δ119905))2

for 119899 = 0

(27)

4 Journal of Applied Mathematics

from where we define

119877119899= 120595 (Δ119905)

119899

sum

119898=0

119903119898 (28)

The previously mentioned framework leads to the followingmain result

Theorem 2 Let 119906 be the solution of (9) and 119880119899

ℎ119873

119899=0in

119881ℎ

a sequence defined by (17)ndash(21) Suppose that 119906 isin

1198712[(0 119879)119867

2(Ω)] 120597119906120597119905 isin 119871

2[(0 119879)119867

2(Ω)] and 120597119896119906120597119905119896 isin

1198712[(0 119879) 119871

2(Ω)] for 119896 = 3 4 Then there exists a constant

119862 gt 0 independent of Δ119905 and the mesh refinement size ℎ

max0le119899le119873

1003817100381710038171003817119906 (sdot Δ119905) minus 119880119899

ℎ

10038171003817100381710038170le 119862 [ℎ

2+ (Δ119905)

2] (29)

Furthermore the discrete solution replicates all the propertiesof solution of the hyperbolic equation in the limiting case of thespace independent equation

We prove the previousTheorem 2 thanks to the followingseries of results

Proposition 3 The following result holds for a constant 119862 gt 0

that is independent of Δ119905 and the mesh refinement size ℎ

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862(100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+120595 (Δ119905) max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(30)

Proof In view of (23) we have the following relation usingtriangular inequality

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

(31)

We bound 120601119899 in (31) by first taking partial sums of the firstterm of (26) seconded by adding the remaining terms from119899 = 1 to119898 for 1 le 119898 le 119873 minus 1 and multiplying both sides by120595(Δ119905) yields

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) minus (

1206011minus 1206010

120595 (Δ119905) Vℎ) + 120595 (Δ119905)

119898

sum

119899=1

119886 (120601119898 Vℎ)

= 120595 (Δ119905)

119898

sum

119899=1

(119903119899 Vℎ)

(32)

In view of (28) we can define

Φ119898= 120595 (Δ119905)

119898

sum

119899=1

120601119898

Φ0= 0

(33)

and using this in (32) we have

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) + 119886 (Φ

119898 Vℎ) = (119877

119898 Vℎ) forallV

ℎisin 119881ℎ

(34)

If we choose Vℎ= 120601119898+1

+ 120601119898isin 119881ℎin (34) and multiply the

result by 120595(Δ119905) this yields10038171003817100381710038171003817120601119898+110038171003817100381710038171003817

2

0minus10038171003817100381710038171206011198981003817100381710038171003817

2

0+ 120595 (Δ119905) 119886 (Φ

119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=1

(119877119898 120601119898+1

+ 120601119898)

for 0 le 119898 le 119873 minus 1

(35)

Summing this from 119898 = 0 to 119898 = 119899 minus 1 for 1 le 119899 le 119873 minus 1

gives

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+1

+ 120601119898)

(36)

Since 119886 is symmetricΦ0 = 0 and

Φ119898+1

minus Φ119898minus1

= 120595 (Δ119905) (120601119898+ 120601119898+1

) 119898 = 1 2 119873 minus 1

(37)

then we deduce in view of the third term of (36)

120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

+ Φ119898minus1

)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

)

minus

119899minus1

sum

119898=0

119886 (Φ119898+1

Φ119898minus1

)

= 119886 (Φ119899minus1

Φ119899)

(38)

By symmetry and coercivity properties of 119886 and the fact that

Φ119899minus Φ119899minus1

= 120595 (Δ119905) 120601119899 for 119899 = 1 2 119873 (39)

we have in view of (38)

(120595 (Δ119905))2

2119886 (120601119899 120601119899)

=1

2119886 (Φ119899minus Φ119899minus119894 Φ119899minus Φ119899minus1

)

=1

2119886 (Φ119899 Φ119899) minus 119886 (Φ

119899 Φ119899minus1

)

+1

2119886 (Φ119899minus1

Φ119899minus1

)

(40)

Journal of Applied Mathematics 5

and so

(120595 (Δ119905))2

2119886 (120601119899 120601119899) ge minus119886 (Φ

119899 Φ119899minus1

) (41)

and this together with (36) yields

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2119886 (120601119899 120601119899)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

for 1 le 119899 le 119873

(42)

By Poincare inequality together with the continuity andcoercivity of 119886 we have the following inequality

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2

10038171003817100381710038171206011198991003817100381710038171003817

2

0

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

(43)

We suppose at this stage that ℎ and Δ119905 satisfy the CFLcondition such that 120595(Δ119905)ℎ lt 1 With this condition theprevious inequality becomes

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

1 le 119899 le 119873

(44)

from where we have 12 lt 119862Δ119905= 1 minus (120595(Δ119905))

22ℎ2lt 1 and

12 lt 119862Δ119905lt 1 By the use of Cauchy-Schwarz inequality on

(44) and some algebraic manipulations we have

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+ 120601119898+1

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

(100381710038171003817100381712060111989810038171003817100381710038170

+10038171003817100381710038171003817120601119898+1100381710038171003817100381710038170

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 2120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

max119898=0119899

100381710038171003817100381712060111989810038171003817100381710038170

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+119862Δ119905

2max119898=1119899

10038171003817100381710038171206011198981003817100381710038171003817

2

0

+2

119862Δ119905

(120595(Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

for 0 le 119898 le 119899

(45)

and since the right-hand side is independent of 119899 then

(119862Δ119905minus119862Δ119905

2)10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(46)

which then implies that

119862Δ119905

2

10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(47)

Furthermore we have

10038171003817100381710038171206011198991003817100381710038171003817

2

0le

2

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0+4(120595 (Δ119905))

2

119862Δ119905

(

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(48)

and hence the following result

max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

le radic2

119862Δ119905

100381710038171003817100381710038171206010100381710038171003817100381710038170

+2120595 (Δ119905)

119862Δ119905

119873minus1

sum

119899=1

100381710038171003817100381711987711989910038171003817100381710038170

(49)

With (49) and the fact that100381710038171003817100381710038171206010100381710038171003817100381710038170

le100381710038171003817100381710038171198900100381710038171003817100381710038170

+100381710038171003817100381710038171205780100381710038171003817100381710038170

(50)

we have in viewof (31) the desired estimate of the proposition

We now proceed to bound the term containing 119877119899 on

the right-hand side of Proposition 3 We achieve this byestimating in the 1198712-norm the function 119903119899 for the cases 119899 = 0

and follow by the case when 119899 ge 1

Lemma 4 There holds100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862[(120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

]

(51)

with a constant 119862 gt 0 that is independent of ℎ and the meshsize

Proof In view of (27) we have 1199030 = (120595(Δ119905))minus2(1206011minus 1206010) We

estimate 1206011 minus 12060100by taking V

ℎisin 119881ℎarbitrary as follows

(1206011minus 1206010 Vℎ)

= (1199081

ℎminus 1198801

ℎ Vℎ) minus (119908

0

ℎminus 1198800

ℎ Vℎ)

= ((120587ℎminus 119868) (119906

1minus 1199060) Vℎ) + (119906

1minus 1198801 Vℎ)

(52)

where we have used (1199060minus1198800

ℎ Vℎ) = (119906

0minus119875ℎ1199060 V) = 0 in view

of (19) It now follows from (52) that1003816100381610038161003816((120587ℎ minus 119868) (1199061 minus 1199060) Vℎ)

1003816100381610038161003816

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

(120597

120597119905(120587ℎminus 119868) 119906 (sdot 119904) V

ℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By Lemma 1

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

((120587ℎminus 119868)

120597119906 (sdot 119904)

120597119905 Vℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By (18)

le 119862120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

1003817100381710038171003817Vℎ10038171003817100381710038170

(53)

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

By the use of the energymethod andGronwallrsquos Lemma thereexists a discrete Galerkin solution 119906

ℎisin 119881ℎsuch that

(1205972119906ℎ

1205971199052 Vℎ) minus 119886 (119906

ℎ Vℎ) = (119891 V

ℎ) forallV

ℎisin 119881ℎ 119905 isin [0 119879]

(119906ℎ Vℎ) = (119875

ℎ1199060 Vℎ)

(120597119906ℎ

120597119905 Vℎ) = (119875

ℎ1199061 Vℎ)

(12)

where 119875ℎdenote the 1198712-projection onto 119881

ℎ Furthermore we

let 119906 isin 1198672(Ω) and we define the Galerkin projection 120587

ℎ119906 of

119906 in 119881ℎby requiring that

119886 (120587ℎ119906 V) = 119886 (119906 V) forallV isin 119881

ℎ (13)

120597119896(120587ℎ119906)

120597119905119896= 120587ℎ(120597119896119906

120597119905119896) 119896 = 0 1 2 119905 isin [0 119879] (14)

The previous essential tools lead to the immediate conse-quence of the approximation properties (11)

Lemma 1 Let 119906 be the solution of (9) Then there existsa unique mapping 120587

ℎ119906 isin 119871

2[(0 119879) 119881

ℎ] which satisfies

(13) Furthermore if for some integer 119896 ge 0 120597119896119906120597119905119896 isin

119871119901[(0 119879)119867

2(Ω)] then

120597119896(120587ℎ119906)

120597119905119896isin 119871119901[(0 119879) 119881

ℎ]

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

[119906 minus 120587ℎ119906]

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198712(Ω)]

le 119862ℎ2

1003817100381710038171003817100381710038171003817100381710038171003817

(120597

120597119905)

119896

119906

1003817100381710038171003817100381710038171003817100381710038171003817119871119901[(0119879)1198672(Ω)]

(15)

for some constant 119862 independent of the mesh size

4 Coupled Nonstandard Finite Difference andContinuous Galerkin Methods

Instead of the continuous Galerkin method summarizedpreviously we present in this section a reliable schemeNSFD-CG consisting of the nonstandard finite difference methodin time and the continuous Galerkin method in the spacevariable We show that the numerical solution obtained fromthe scheme NSFD-CG attained the optimal convergence inboth the energy and 119871

2-norms We proceed in this regardby letting the step size 119905

119899= 119899Δ119905 for 119899 = 0 1 2 119873 For a

sufficiently smooth function V(119909 119905) we set

(120597

120597119905)

119896

V119899= (

120597

120597119905)

119896

V (sdot 119905119899) V

119899= V (sdot 119905

119899) 119896 ge 0 (16)

With this we proceed to find the NSFD-CG approximation119880119899

ℎ such that 119880119899

ℎasymp 119906119899

ℎat discrete time 119905

119899 That is find a

sequence 119880119899ℎ119873

119899=0in 119881ℎsuch that

(1205752119880119899

ℎ Vℎ) + 119886 (119880

119899

ℎ Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(17)

where

1205752119880119899

ℎ=119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

(120595 (Δ119905))2

119899 = 1 2 119873 minus 1 (18)

and 120595(Δ119905) = 2 sin(Δ1199052) restricted between 0 lt 120595(Δ119905) lt 1The initial conditions 1198800

ℎisin 119881ℎand 1198801

ℎisin 119881ℎare given by

(1198800

ℎ Vℎ) = (119875

ℎ1199060 Vℎ) (19)

(1198801

ℎ Vℎ) = (119880

0

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ119905))

2

20

ℎ Vℎ)

forallVℎisin 119881ℎ

(20)

where 0ℎisin 119881ℎis defined by

(0

ℎ Vℎ) = (119891

0 Vℎ) minus 119886 (119906

0 Vℎ) forallV

ℎisin 119881ℎ (21)

If 119891 = 0 in (1) we will have in view of (17) an exact scheme

(119880119899+1

ℎminus 2119880119899

ℎ+ 119880119899minus1

ℎ

4sin2 (Δ1199052) Vℎ) + (nabla119880

119899

ℎ nablaVℎ) = 0 (22)

which according to Mickens [12] replicates both the energypreserving features and the properties of the exact solution(1)ndash(4) In order to state and prove the main result we needa framework on which this result is based To this end weproceed by decomposing the error denoted by 119890119899 = 119906

119899minus 119880119899

ℎ

into the following error equation

119890119899= 119906119899minus 119908119899

ℎ+ 119908119899

ℎminus 119880119899

ℎ= 120578119899minus 120601119899 (23)

where 119908119899ℎ= 120587ℎ119906119899isin 119881ℎis the Galerkin projector of 119906119899

Due to the regularity assumptionmentioned earlier the exactsolution 119906 of (1)ndash(4) satisfies

(1205972119906119899

1205971199052 Vℎ) + 119886 (119906

119899 Vℎ) = (119891

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873

(24)

Subtracting (17) from (24) and using some properties of theGalerkin projection in the space we have

(1205752119908119899

ℎminus 1205752119880119899

ℎ Vℎ) + 119886 (119908

119899

ℎminus 119906119899 Vℎ)

= (1205752119908119899

ℎminus1205972119906119899

1205971199052 Vℎ)

(25)

from where we obtain in view of (23)

(1205752120601119899 Vℎ) + (120601

119899 Vℎ) = (119903

119899 Vℎ)

forallVℎisin 119881ℎ 119899 = 1 2 119873 minus 1

(26)

where 119903119899 = 1205752119908119899

ℎminus12059721199061198991205971199052 In view of the necessity for error

bound we set

119903119899=

1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1

1206011minus 1206010

(120595 (Δ119905))2

for 119899 = 0

(27)

4 Journal of Applied Mathematics

from where we define

119877119899= 120595 (Δ119905)

119899

sum

119898=0

119903119898 (28)

The previously mentioned framework leads to the followingmain result

Theorem 2 Let 119906 be the solution of (9) and 119880119899

ℎ119873

119899=0in

119881ℎ

a sequence defined by (17)ndash(21) Suppose that 119906 isin

1198712[(0 119879)119867

2(Ω)] 120597119906120597119905 isin 119871

2[(0 119879)119867

2(Ω)] and 120597119896119906120597119905119896 isin

1198712[(0 119879) 119871

2(Ω)] for 119896 = 3 4 Then there exists a constant

119862 gt 0 independent of Δ119905 and the mesh refinement size ℎ

max0le119899le119873

1003817100381710038171003817119906 (sdot Δ119905) minus 119880119899

ℎ

10038171003817100381710038170le 119862 [ℎ

2+ (Δ119905)

2] (29)

Furthermore the discrete solution replicates all the propertiesof solution of the hyperbolic equation in the limiting case of thespace independent equation

We prove the previousTheorem 2 thanks to the followingseries of results

Proposition 3 The following result holds for a constant 119862 gt 0

that is independent of Δ119905 and the mesh refinement size ℎ

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862(100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+120595 (Δ119905) max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(30)

Proof In view of (23) we have the following relation usingtriangular inequality

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

(31)

We bound 120601119899 in (31) by first taking partial sums of the firstterm of (26) seconded by adding the remaining terms from119899 = 1 to119898 for 1 le 119898 le 119873 minus 1 and multiplying both sides by120595(Δ119905) yields

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) minus (

1206011minus 1206010

120595 (Δ119905) Vℎ) + 120595 (Δ119905)

119898

sum

119899=1

119886 (120601119898 Vℎ)

= 120595 (Δ119905)

119898

sum

119899=1

(119903119899 Vℎ)

(32)

In view of (28) we can define

Φ119898= 120595 (Δ119905)

119898

sum

119899=1

120601119898

Φ0= 0

(33)

and using this in (32) we have

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) + 119886 (Φ

119898 Vℎ) = (119877

119898 Vℎ) forallV

ℎisin 119881ℎ

(34)

If we choose Vℎ= 120601119898+1

+ 120601119898isin 119881ℎin (34) and multiply the

result by 120595(Δ119905) this yields10038171003817100381710038171003817120601119898+110038171003817100381710038171003817

2

0minus10038171003817100381710038171206011198981003817100381710038171003817

2

0+ 120595 (Δ119905) 119886 (Φ

119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=1

(119877119898 120601119898+1

+ 120601119898)

for 0 le 119898 le 119873 minus 1

(35)

Summing this from 119898 = 0 to 119898 = 119899 minus 1 for 1 le 119899 le 119873 minus 1

gives

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+1

+ 120601119898)

(36)

Since 119886 is symmetricΦ0 = 0 and

Φ119898+1

minus Φ119898minus1

= 120595 (Δ119905) (120601119898+ 120601119898+1

) 119898 = 1 2 119873 minus 1

(37)

then we deduce in view of the third term of (36)

120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

+ Φ119898minus1

)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

)

minus

119899minus1

sum

119898=0

119886 (Φ119898+1

Φ119898minus1

)

= 119886 (Φ119899minus1

Φ119899)

(38)

By symmetry and coercivity properties of 119886 and the fact that

Φ119899minus Φ119899minus1

= 120595 (Δ119905) 120601119899 for 119899 = 1 2 119873 (39)

we have in view of (38)

(120595 (Δ119905))2

2119886 (120601119899 120601119899)

=1

2119886 (Φ119899minus Φ119899minus119894 Φ119899minus Φ119899minus1

)

=1

2119886 (Φ119899 Φ119899) minus 119886 (Φ

119899 Φ119899minus1

)

+1

2119886 (Φ119899minus1

Φ119899minus1

)

(40)

Journal of Applied Mathematics 5

and so

(120595 (Δ119905))2

2119886 (120601119899 120601119899) ge minus119886 (Φ

119899 Φ119899minus1

) (41)

and this together with (36) yields

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2119886 (120601119899 120601119899)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

for 1 le 119899 le 119873

(42)

By Poincare inequality together with the continuity andcoercivity of 119886 we have the following inequality

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2

10038171003817100381710038171206011198991003817100381710038171003817

2

0

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

(43)

We suppose at this stage that ℎ and Δ119905 satisfy the CFLcondition such that 120595(Δ119905)ℎ lt 1 With this condition theprevious inequality becomes

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

1 le 119899 le 119873

(44)

from where we have 12 lt 119862Δ119905= 1 minus (120595(Δ119905))

22ℎ2lt 1 and

12 lt 119862Δ119905lt 1 By the use of Cauchy-Schwarz inequality on

(44) and some algebraic manipulations we have

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+ 120601119898+1

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

(100381710038171003817100381712060111989810038171003817100381710038170

+10038171003817100381710038171003817120601119898+1100381710038171003817100381710038170

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 2120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

max119898=0119899

100381710038171003817100381712060111989810038171003817100381710038170

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+119862Δ119905

2max119898=1119899

10038171003817100381710038171206011198981003817100381710038171003817

2

0

+2

119862Δ119905

(120595(Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

for 0 le 119898 le 119899

(45)

and since the right-hand side is independent of 119899 then

(119862Δ119905minus119862Δ119905

2)10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(46)

which then implies that

119862Δ119905

2

10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(47)

Furthermore we have

10038171003817100381710038171206011198991003817100381710038171003817

2

0le

2

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0+4(120595 (Δ119905))

2

119862Δ119905

(

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(48)

and hence the following result

max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

le radic2

119862Δ119905

100381710038171003817100381710038171206010100381710038171003817100381710038170

+2120595 (Δ119905)

119862Δ119905

119873minus1

sum

119899=1

100381710038171003817100381711987711989910038171003817100381710038170

(49)

With (49) and the fact that100381710038171003817100381710038171206010100381710038171003817100381710038170

le100381710038171003817100381710038171198900100381710038171003817100381710038170

+100381710038171003817100381710038171205780100381710038171003817100381710038170

(50)

we have in viewof (31) the desired estimate of the proposition

We now proceed to bound the term containing 119877119899 on

the right-hand side of Proposition 3 We achieve this byestimating in the 1198712-norm the function 119903119899 for the cases 119899 = 0

and follow by the case when 119899 ge 1

Lemma 4 There holds100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862[(120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

]

(51)

with a constant 119862 gt 0 that is independent of ℎ and the meshsize

Proof In view of (27) we have 1199030 = (120595(Δ119905))minus2(1206011minus 1206010) We

estimate 1206011 minus 12060100by taking V

ℎisin 119881ℎarbitrary as follows

(1206011minus 1206010 Vℎ)

= (1199081

ℎminus 1198801

ℎ Vℎ) minus (119908

0

ℎminus 1198800

ℎ Vℎ)

= ((120587ℎminus 119868) (119906

1minus 1199060) Vℎ) + (119906

1minus 1198801 Vℎ)

(52)

where we have used (1199060minus1198800

ℎ Vℎ) = (119906

0minus119875ℎ1199060 V) = 0 in view

of (19) It now follows from (52) that1003816100381610038161003816((120587ℎ minus 119868) (1199061 minus 1199060) Vℎ)

1003816100381610038161003816

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

(120597

120597119905(120587ℎminus 119868) 119906 (sdot 119904) V

ℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By Lemma 1

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

((120587ℎminus 119868)

120597119906 (sdot 119904)

120597119905 Vℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By (18)

le 119862120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

1003817100381710038171003817Vℎ10038171003817100381710038170

(53)

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

from where we define

119877119899= 120595 (Δ119905)

119899

sum

119898=0

119903119898 (28)

The previously mentioned framework leads to the followingmain result

Theorem 2 Let 119906 be the solution of (9) and 119880119899

ℎ119873

119899=0in

119881ℎ

a sequence defined by (17)ndash(21) Suppose that 119906 isin

1198712[(0 119879)119867

2(Ω)] 120597119906120597119905 isin 119871

2[(0 119879)119867

2(Ω)] and 120597119896119906120597119905119896 isin

1198712[(0 119879) 119871

2(Ω)] for 119896 = 3 4 Then there exists a constant

119862 gt 0 independent of Δ119905 and the mesh refinement size ℎ

max0le119899le119873

1003817100381710038171003817119906 (sdot Δ119905) minus 119880119899

ℎ

10038171003817100381710038170le 119862 [ℎ

2+ (Δ119905)

2] (29)

Furthermore the discrete solution replicates all the propertiesof solution of the hyperbolic equation in the limiting case of thespace independent equation

We prove the previousTheorem 2 thanks to the followingseries of results

Proposition 3 The following result holds for a constant 119862 gt 0

that is independent of Δ119905 and the mesh refinement size ℎ

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862(100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+120595 (Δ119905) max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(30)

Proof In view of (23) we have the following relation usingtriangular inequality

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

(31)

We bound 120601119899 in (31) by first taking partial sums of the firstterm of (26) seconded by adding the remaining terms from119899 = 1 to119898 for 1 le 119898 le 119873 minus 1 and multiplying both sides by120595(Δ119905) yields

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) minus (

1206011minus 1206010

120595 (Δ119905) Vℎ) + 120595 (Δ119905)

119898

sum

119899=1

119886 (120601119898 Vℎ)

= 120595 (Δ119905)

119898

sum

119899=1

(119903119899 Vℎ)

(32)

In view of (28) we can define

Φ119898= 120595 (Δ119905)

119898

sum

119899=1

120601119898

Φ0= 0

(33)

and using this in (32) we have

(120601119898+1

minus 120601119898

120595 (Δ119905) Vℎ) + 119886 (Φ

119898 Vℎ) = (119877

119898 Vℎ) forallV

ℎisin 119881ℎ

(34)

If we choose Vℎ= 120601119898+1

+ 120601119898isin 119881ℎin (34) and multiply the

result by 120595(Δ119905) this yields10038171003817100381710038171003817120601119898+110038171003817100381710038171003817

2

0minus10038171003817100381710038171206011198981003817100381710038171003817

2

0+ 120595 (Δ119905) 119886 (Φ

119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=1

(119877119898 120601119898+1

+ 120601119898)

for 0 le 119898 le 119873 minus 1

(35)

Summing this from 119898 = 0 to 119898 = 119899 minus 1 for 1 le 119899 le 119873 minus 1

gives

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

= 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+1

+ 120601119898)

(36)

Since 119886 is symmetricΦ0 = 0 and

Φ119898+1

minus Φ119898minus1

= 120595 (Δ119905) (120601119898+ 120601119898+1

) 119898 = 1 2 119873 minus 1

(37)

then we deduce in view of the third term of (36)

120595 (Δ119905)

119899minus1

sum

119898=0

119886 (Φ119898 120601119898+1

+ 120601119898)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

+ Φ119898minus1

)

=

119899minus1

sum

119898=0

119886 (Φ119898 Φ119898+1

)

minus

119899minus1

sum

119898=0

119886 (Φ119898+1

Φ119898minus1

)

= 119886 (Φ119899minus1

Φ119899)

(38)

By symmetry and coercivity properties of 119886 and the fact that

Φ119899minus Φ119899minus1

= 120595 (Δ119905) 120601119899 for 119899 = 1 2 119873 (39)

we have in view of (38)

(120595 (Δ119905))2

2119886 (120601119899 120601119899)

=1

2119886 (Φ119899minus Φ119899minus119894 Φ119899minus Φ119899minus1

)

=1

2119886 (Φ119899 Φ119899) minus 119886 (Φ

119899 Φ119899minus1

)

+1

2119886 (Φ119899minus1

Φ119899minus1

)

(40)

Journal of Applied Mathematics 5

and so

(120595 (Δ119905))2

2119886 (120601119899 120601119899) ge minus119886 (Φ

119899 Φ119899minus1

) (41)

and this together with (36) yields

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2119886 (120601119899 120601119899)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

for 1 le 119899 le 119873

(42)

By Poincare inequality together with the continuity andcoercivity of 119886 we have the following inequality

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2

10038171003817100381710038171206011198991003817100381710038171003817

2

0

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

(43)

We suppose at this stage that ℎ and Δ119905 satisfy the CFLcondition such that 120595(Δ119905)ℎ lt 1 With this condition theprevious inequality becomes

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

1 le 119899 le 119873

(44)

from where we have 12 lt 119862Δ119905= 1 minus (120595(Δ119905))

22ℎ2lt 1 and

12 lt 119862Δ119905lt 1 By the use of Cauchy-Schwarz inequality on

(44) and some algebraic manipulations we have

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+ 120601119898+1

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

(100381710038171003817100381712060111989810038171003817100381710038170

+10038171003817100381710038171003817120601119898+1100381710038171003817100381710038170

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 2120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

max119898=0119899

100381710038171003817100381712060111989810038171003817100381710038170

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+119862Δ119905

2max119898=1119899

10038171003817100381710038171206011198981003817100381710038171003817

2

0

+2

119862Δ119905

(120595(Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

for 0 le 119898 le 119899

(45)

and since the right-hand side is independent of 119899 then

(119862Δ119905minus119862Δ119905

2)10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(46)

which then implies that

119862Δ119905

2

10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(47)

Furthermore we have

10038171003817100381710038171206011198991003817100381710038171003817

2

0le

2

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0+4(120595 (Δ119905))

2

119862Δ119905

(

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(48)

and hence the following result

max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

le radic2

119862Δ119905

100381710038171003817100381710038171206010100381710038171003817100381710038170

+2120595 (Δ119905)

119862Δ119905

119873minus1

sum

119899=1

100381710038171003817100381711987711989910038171003817100381710038170

(49)

With (49) and the fact that100381710038171003817100381710038171206010100381710038171003817100381710038170

le100381710038171003817100381710038171198900100381710038171003817100381710038170

+100381710038171003817100381710038171205780100381710038171003817100381710038170

(50)

we have in viewof (31) the desired estimate of the proposition

We now proceed to bound the term containing 119877119899 on

the right-hand side of Proposition 3 We achieve this byestimating in the 1198712-norm the function 119903119899 for the cases 119899 = 0

and follow by the case when 119899 ge 1

Lemma 4 There holds100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862[(120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

]

(51)

with a constant 119862 gt 0 that is independent of ℎ and the meshsize

Proof In view of (27) we have 1199030 = (120595(Δ119905))minus2(1206011minus 1206010) We

estimate 1206011 minus 12060100by taking V

ℎisin 119881ℎarbitrary as follows

(1206011minus 1206010 Vℎ)

= (1199081

ℎminus 1198801

ℎ Vℎ) minus (119908

0

ℎminus 1198800

ℎ Vℎ)

= ((120587ℎminus 119868) (119906

1minus 1199060) Vℎ) + (119906

1minus 1198801 Vℎ)

(52)

where we have used (1199060minus1198800

ℎ Vℎ) = (119906

0minus119875ℎ1199060 V) = 0 in view

of (19) It now follows from (52) that1003816100381610038161003816((120587ℎ minus 119868) (1199061 minus 1199060) Vℎ)

1003816100381610038161003816

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

(120597

120597119905(120587ℎminus 119868) 119906 (sdot 119904) V

ℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By Lemma 1

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

((120587ℎminus 119868)

120597119906 (sdot 119904)

120597119905 Vℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By (18)

le 119862120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

1003817100381710038171003817Vℎ10038171003817100381710038170

(53)

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

and so

(120595 (Δ119905))2

2119886 (120601119899 120601119899) ge minus119886 (Φ

119899 Φ119899minus1

) (41)

and this together with (36) yields

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2119886 (120601119899 120601119899)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

for 1 le 119899 le 119873

(42)

By Poincare inequality together with the continuity andcoercivity of 119886 we have the following inequality

10038171003817100381710038171206011198991003817100381710038171003817

2

0minus(120595 (Δ119905))

2

2

10038171003817100381710038171206011198991003817100381710038171003817

2

0

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

(43)

We suppose at this stage that ℎ and Δ119905 satisfy the CFLcondition such that 120595(Δ119905)ℎ lt 1 With this condition theprevious inequality becomes

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119899 120601119898+ 120601119898+1

)

1 le 119899 le 119873

(44)

from where we have 12 lt 119862Δ119905= 1 minus (120595(Δ119905))

22ℎ2lt 1 and

12 lt 119862Δ119905lt 1 By the use of Cauchy-Schwarz inequality on

(44) and some algebraic manipulations we have

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

(119877119898 120601119898+ 120601119898+1

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

(100381710038171003817100381712060111989810038171003817100381710038170

+10038171003817100381710038171003817120601119898+1100381710038171003817100381710038170

)

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+ 2120595 (Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989810038171003817100381710038170

max119898=0119899

100381710038171003817100381712060111989810038171003817100381710038170

le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+119862Δ119905

2max119898=1119899

10038171003817100381710038171206011198981003817100381710038171003817

2

0

+2

119862Δ119905

(120595(Δ119905)

119899minus1

sum

119898=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

for 0 le 119898 le 119899

(45)

and since the right-hand side is independent of 119899 then

(119862Δ119905minus119862Δ119905

2)10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(46)

which then implies that

119862Δ119905

2

10038171003817100381710038171206011198981003817100381710038171003817

2

0le10038171003817100381710038171003817120601010038171003817100381710038171003817

2

0+

2

119862Δ119905

(120595(Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(47)

Furthermore we have

10038171003817100381710038171206011198991003817100381710038171003817

2

0le

2

119862Δ119905

10038171003817100381710038171206011198991003817100381710038171003817

2

0+4(120595 (Δ119905))

2

119862Δ119905

(

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

)

2

(48)

and hence the following result

max119899=0119873

100381710038171003817100381712060111989910038171003817100381710038170

le radic2

119862Δ119905

100381710038171003817100381710038171206010100381710038171003817100381710038170

+2120595 (Δ119905)

119862Δ119905

119873minus1

sum

119899=1

100381710038171003817100381711987711989910038171003817100381710038170

(49)

With (49) and the fact that100381710038171003817100381710038171206010100381710038171003817100381710038170

le100381710038171003817100381710038171198900100381710038171003817100381710038170

+100381710038171003817100381710038171205780100381710038171003817100381710038170

(50)

we have in viewof (31) the desired estimate of the proposition

We now proceed to bound the term containing 119877119899 on

the right-hand side of Proposition 3 We achieve this byestimating in the 1198712-norm the function 119903119899 for the cases 119899 = 0

and follow by the case when 119899 ge 1

Lemma 4 There holds100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862[(120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

]

(51)

with a constant 119862 gt 0 that is independent of ℎ and the meshsize

Proof In view of (27) we have 1199030 = (120595(Δ119905))minus2(1206011minus 1206010) We

estimate 1206011 minus 12060100by taking V

ℎisin 119881ℎarbitrary as follows

(1206011minus 1206010 Vℎ)

= (1199081

ℎminus 1198801

ℎ Vℎ) minus (119908

0

ℎminus 1198800

ℎ Vℎ)

= ((120587ℎminus 119868) (119906

1minus 1199060) Vℎ) + (119906

1minus 1198801 Vℎ)

(52)

where we have used (1199060minus1198800

ℎ Vℎ) = (119906

0minus119875ℎ1199060 V) = 0 in view

of (19) It now follows from (52) that1003816100381610038161003816((120587ℎ minus 119868) (1199061 minus 1199060) Vℎ)

1003816100381610038161003816

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

(120597

120597119905(120587ℎminus 119868) 119906 (sdot 119904) V

ℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By Lemma 1

le int

1199051

0

10038161003816100381610038161003816100381610038161003816

((120587ℎminus 119868)

120597119906 (sdot 119904)

120597119905 Vℎ)

10038161003816100381610038161003816100381610038161003816

119889119904 By (18)

le 119862120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

1003817100381710038171003817Vℎ10038171003817100381710038170

(53)

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

We now proceed to estimate the term (1199061minus 1198801

ℎ V) in (52)

as follows by Taylorrsquos formula and the fact 119906(sdot 0) = 1199060and

120597119906(sdot 0)120597119905 = 1199061in (1)ndash(3) we have

1199061= 1199060+ 120595 (Δ119905) 119906

1+(120595 (Δ119905))

2

2

1205972119906 (sdot 0)

1205971199052

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2 1205973119906 (sdot 119904)

1205971199053119889119904

(54)

In view of the definition of 1198801ℎin (20) and the fact that

(1199060minus 119875ℎ1199060 V) = 0 (119906

1minus 119875ℎ1199061 V) = 0 (55)

we have

1198801

ℎ= 1198800

ℎ+ 120595 (Δ119905) 119875

ℎ1199061+(120595 (Δ))

2

20

ℎ (56)

We then deduce from (54) and (56) that

(1199061minus 1198801

ℎ Vℎ) =

(120595 (Δ119905))2

2(1205972119906 (sdot 0)

1205971199052minus 0

ℎ Vℎ)

+1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

(1205973119906 (sdot 119904)

1205971199053 Vℎ)119889119904

(57)

But by the definition of 0ℎin (20) we have in view of (21) that

(1205972119906 (sdot 119904)

1205971199052minus 0

ℎ Vℎ)

= (1198910 Vℎ) minus 119886 (119906

0 Vℎ) minus (119891

0 Vℎ) + 119886 (119906

0 Vℎ) = 0

(58)

implying that10038161003816100381610038161003816(1199061minus 1198801

ℎ V)

10038161003816100381610038161003816

le1

2int

1199051

0

(120595 (Δ119905) minus 119904)2

100381610038161003816100381610038161003816100381610038161003816

(1205973119906 (sdot 119904)

1205971199053 V)

100381610038161003816100381610038161003816100381610038161003816

119889119904

le 119862(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

V0

(59)

Since1206011minus1206010 isin 119881ℎ then (52) together with (53) and (59) yields

100381710038171003817100381710038171206011minus 1206010100381710038171003817100381710038170

le 119862(120595 (Δ119905) ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(60)

and dividing throughout by (120595(Δ119905))2 yields

100381710038171003817100381710038171199030100381710038171003817100381710038170

le 119862((120595 (Δ119905))minus1

ℎ2

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+120595 (Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(61)

as required

Lemma 5 For 1 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711990311989910038171003817100381710038170

le 119862(ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906 (sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

+120595 (Δ119905) int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904)

(62)

with a constant 119862 gt 0 that is independent of ℎ and mesh size

Proof In view of (27) we have

119903119899= 1205752119908119899

ℎminus1205972119906119899

1205971199052 for 119899 = 1 2 119873 minus 1 (63)

By the triangular inequality we have

100381710038171003817100381711990311989910038171003817100381710038170

=

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le100381710038171003817100381710038171205752(120587ℎminus 119868)119906119899100381710038171003817100381710038170

+

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

(64)

and using the following identityV (sdot 119905119899+1

) minus 2V (sdot 119905119899) + V (sdot 119905

119899minus1)

= Δ119905int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)1205972V (sdot 119904)

1205971199052119889119904

(65)

on the first term of the right-hand side of (64) we have100381710038171003817100381710038171205752(120587ℎminus 119868) 119906

119899100381710038171003817100381710038170

le1

120595 (Δ119905)int

119905119899+1

119905119899minus1

(1 minus

1003816100381610038161003816119904 minus 1199051198991003816100381610038161003816

Δ119905)

100381710038171003817100381710038171003817100381710038171003817

1205972(120587ℎminus 119868)119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le 119862ℎ2

120595 (Δ119905)int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205972119906(sdot 119904)

1205971199052

1003817100381710038171003817100381710038171003817100381710038172

119889119904

(66)

after the use of Lemma 1 and (14)The second term of (64) can be estimated by the use of

the identity

1205752119908119899

ℎminus1205972119906119899

1205971199052

=1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3 1205974119906 (sdot 119904)

1205971199054119889119904

(67)

which is obtained from Taylorrsquos formulae with integralremainder This is deduced as follows

100381710038171003817100381710038171003817100381710038171003817

1205752119908119899

ℎminus1205972119906119899

1205971199052

1003817100381710038171003817100381710038171003817100381710038170

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(Δ119905 minus1003816100381610038161003816119904 minus 119905119899

1003816100381610038161003816)3

100381710038171003817100381710038171003817100381710038171003817

1205974119906 (sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le1

6(Δ119905)2int

119905119899+1

119905119899minus1

(120595 (Δ119905))3

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

le120595 (Δ119905)

6int

119905119899+1

119905119899minus1

100381710038171003817100381710038171003817100381710038171003817

1205974119906(sdot 119904)

1205971199054

1003817100381710038171003817100381710038171003817100381710038170

119889119904

(68)

using the relation that 120595(Δ119905) minus |119904 minus 119905119899| le 120595(Δ119905) in (68)

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

In view of (64) using (66) and (68) we have the desiredresult for the bound of 119903119899

0Wenow assemble Lemmas 4 and

5 in the next proposition to obtain the bound 119877119899 as follows

Proposition 6 For 0 le 119899 le 119873 minus 1 there holds

100381710038171003817100381711987711989910038171003817100381710038170

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(69)

Proof Using the bounds on 1199031198990in Lemmas 4 and 5 we have

100381710038171003817100381711987711989910038171003817100381710038170

le 120595 (Δ119905)100381710038171003817100381710038171199030100381710038171003817100381710038170

+ 120595 (Δ119905)

119873minus1

sum

119898=1

119903119898

le 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

+ 119862ℎ2(

10038171003817100381710038171003817100381710038171003817

120597119906

120597119905

100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205972119906

1205971199052

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198672(Ω)]

)

(70)

and the proof is completed

With all these results we are now in the position to provethe main result as follows

Proof of Theorem 2 Using Proposition 3 and the fact that

120595 (Δ119905)

119873minus1

sum

119899=0

100381710038171003817100381711987711989910038171003817100381710038170

le 119879 max119899=0119873

100381710038171003817100381711987711989910038171003817100381710038170

(71)

we have

max119899=0119873

10038171003817100381710038171198901198991003817100381710038171003817 le 119862(

100381710038171003817100381710038171198900100381710038171003817100381710038170

+ max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

+ 119879 max119899=0119873minus1

100381710038171003817100381711987711989910038171003817100381710038170

)

(72)

By the use of Lemma 1 we can bound the second term on theright-hand side as follows

max119899=0119873

100381710038171003817100381712057811989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)] (73)

From the approximation property of the 1198712-projection wehave

100381710038171003817100381710038171198900100381710038171003817100381710038170

le 119862ℎ210038171003817100381710038171199060

10038171003817100381710038172le 119862ℎ21199061198712[(0119879)1198672(Ω)] (74)

Finally by the bound of max119899=0119873minus1

1198771198990obtained via

Proposition 6 we have the result

max119899=0119873

100381710038171003817100381711989011989910038171003817100381710038170

le 119862ℎ21199061198712[(0119879)1198672(Ω)]

+ 119862(120595 (Δ119905))2

(

100381710038171003817100381710038171003817100381710038171003817

1205973119906

1205971199053

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

+

100381710038171003817100381710038171003817100381710038171003817

1205974119906

1205971199054

1003817100381710038171003817100381710038171003817100381710038171198712[(0119879)1198712(Ω)]

)

(75)

In view of the relationship

120595 (Δ119905) = 2 sin(Δ1199052) asymp Δ119905 + 0 [(Δ119905)

2] (76)

as proposed for such schemes in Mickens [12] we havethe required estimate as Δ119905 rarr 0 Furthermore using thefact that its uniform convergence results imply pointwiseconvergence for 119909 isin Ω1015840 completes the proof

5 Numerical Experiments

In this section we present the numerical experiments onproblem (1) using both the standard finite difference (SFD)and NSFD-CG methods These experiments are performedin Ω = (0 1)

2times (0 119879) where Ω was discretized using regular

meshes of sizes ℎ = 1119872 in the space and Δ119905 = 119879119873 inthe time The 119872 and 119873 in such a discretization denote thenumber of nodes and time respectively The initial data wasconsidered to be 119906

0(119909) = 119909

11199092(1 minus 119909

1)(1 minus 119909

2) and 119906

1(119909) = 0

where these data were deduced from the exact solution

119906 (119909 119905) = 11990911199092(1 minus 119909

1) (1 minus 119909

2) cos (120587119905) (77)

Using the previous exact solution we obtained the right-handside 119891 of (1) In the computation (1) together with (17)ndash(20)led to a system of equations

AX = b (78)

In solving for X in the above system we took the followingvalues of 119872 = 10 15 20 25 50 and 100 For a fix 119872 = 20119873 = 10 and 119879 = 1 we had Figures 1ndash3 illustrating varioussolutions corresponding to their respective schemes

Figure 1 shows the exact solution Figure 2 the solutionfrom the SFD-CG and Figure 3 the solution from the NSFD-CG schemes followed by Tables 1 and 2 which demonstratevarious optimal rates of convergence in both 119867

1 and 1198712-

norms of these schemesThese optimal rates of convergence were calculated by

using the formula

Rate =ln (11989021198901)

ln (ℎ2ℎ1) (79)

where ℎ1and ℎ2together with 119890

1and 1198902are successive triangle

diameters and errors respectively These results are self-explanatory and we could conclude that the results as shownby these experiments exhibit the desired results as expectedfrom our theoretical analysis

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 0204 06

081

0

05

10

05

1

15

Exact solution

Figure 1 The Exact solution

002

0406

081

0

05

10

05

1

15

Approximate solution

Figure 2 Approximate solution for SFD-CG scheme

6 Conclusion

We presented a reliable scheme of the wave equation con-sisting of the nonstandard finite difference method in timeand the continuous Galerkin method in the space variable(NSFD-CG) We proved theoretically that the numericalsolution obtained from this scheme attains the optimalrate of convergence in both the energy and the 1198712-normsFurthermore we showed that the scheme under investigationreplicates the properties of the exact solution of the waveequation We proceeded by the help of a numerical exampleand showed that the optimal rate of convergence as provedtheoretically is guaranteed in both the energy and the 1198712-norms This convergence results hold for any fully discreteNSFD-CG method where the scheme under considerationhas a bilinear form which is symmetric continuous andcoercive

The method presented in this paper can be extendedto the nonlinear hyperbolic or parabolic problems witheither smooth or nonsmooth domain if at all these casesfollowed the procedure as proposed by Mickens [12] We will

002

04 0608

1

0

05

10

05

1

15

Approximate solution

Figure 3 Approximate solution for NSFD-CG scheme

Table 1 NSFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3800119864 minus 3 167119864 minus 215 1700119864 minus 3 1110119864 minus 2 1983 1007320 1000119864 minus 3 8200119864 minus 3 18445 0968825 6591119864 minus 4 6700119864 minus 3 18678 1013450 2012119864 minus 4 3400119864 minus 3 17119 09786100 5098119864 minus 5 1800119864 minus 3 19806 09175

Table 2 SFD method error in both 1198712 and1198671-norms

119872 1198712-error 119867

1-error Rate 1198712 Rate1198671

10 3400119864 minus 3 1700119864 minus 215 1700119864 minus 3 1130119864 minus 2 1709 100720 1050119864 minus 3 8500119864 minus 3 1674 098925 1600119864 minus 3 6800119864 minus 3 1887 100050 5170119864 minus 4 3500119864 minus 3 1629 0958100 1799119864 minus 4 1900119864 minus 3 1522 0881

also exploit another form of nonstandard finite differentialmethod as proposed in [18 19]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Big thanks goes to the Almighty God from whose strengthI draw my inspiration to complete this project The researchcontained in this paper has been supported by the Universityof Pretoria South AfricaThe authors would like to thank thereviewers who spent a lot of time to go through the paperThanks also go to the following colleagues Dr ChapwanyaDr Appadu Mr Mbehou and Mr Aderogba who helpchecked the revised version of the work

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

References

[1] D A French and J W Schaeffer ldquoContinuous finite elementmethods which preserve energy properties for nonlinear prob-lemsrdquo Applied Mathematics and Computation vol 39 no 3 pp271ndash295 1990

[2] D A French and T E Peterson ldquoA continuous space-timefinite element method for the wave equationrdquo Mathematics ofComputation vol 65 no 214 pp 491ndash506 1996

[3] R T Glassey ldquoConvergence of an energy-preserving scheme forthe Zakharov equations in one space dimensionrdquo Mathematicsof Computation vol 58 no 197 pp 83ndash102 1992

[4] R Glassey and J Schaeffer ldquoConvergence of a second-orderscheme for semilinear hyperbolic equations in 2 + 1 dimen-sionsrdquoMathematics of Computation vol 56 no 193 pp 87ndash1061991

[5] W Strauss and L Vazquez ldquoNumerical solution of a nonlinearKlein-Gordon equationrdquo Journal of Computational Physics vol28 no 2 pp 271ndash278 1978

[6] TDupont ldquo1198712-estimates forGalerkinmethods for secondorderhyperbolic equationsrdquo SIAM Journal onNumerical Analysis vol10 pp 880ndash889 1973

[7] G A Baker ldquoError estimates for finite element methods forsecond order hyperbolic equationsrdquo SIAM Journal onNumericalAnalysis vol 13 no 4 pp 564ndash576 1976

[8] E Gekeler ldquoLinearmultistepmethods andGalerkin proceduresfor initial boundary value problemsrdquo SIAM Journal on Numeri-cal Analysis vol 13 no 4 pp 536ndash548 1976

[9] C Johnson ldquoDiscontinuous Galerkin finite element methodsfor second order hyperbolic problemsrdquo Computer Methods inApplied Mechanics and Engineering vol 107 no 1-2 pp 117ndash1291993

[10] G R Richter ldquoAn explicit finite element method for thewave equationrdquo Applied Numerical Mathematics vol 16 no1-2 pp 65ndash80 1994 A Festschrift to honor Professor RobertVichnevetsky on his 65th birthday

[11] P W M Chin J K Djoko and J M-S Lubuma ldquoReliablenumerical schemes for a linear diffusion equation on a nons-mooth domainrdquo Applied Mathematics Letters vol 23 no 5 pp544ndash548 2010

[12] R E Mickens Nonstandard Finite Difference Models of Differ-ential Equations World Scientific Publishing River Edge NJUSA 1994

[13] R Anguelov and J M-S Lubuma ldquoContributions to themathematics of the nonstandard finite difference method andapplicationsrdquo Numerical Methods for Partial Differential Equa-tions vol 17 no 5 pp 518ndash543 2001

[14] R Anguelov and J M-S Lubuma ldquoNonstandard finite dif-ference method by nonlocal approximationrdquo Mathematics andComputers in Simulation vol 61 no 3ndash6 pp 465ndash475 2003

[15] S M Moghadas M E Alexander B D Corbett and A BGumel ldquoA positivity-preserving Mickens-type discretizationof an epidemic modelrdquo Journal of Difference Equations andApplications vol 9 no 11 pp 1037ndash1051 2003

[16] K C Patidar ldquoOn the use of nonstandard finite differencemethodsrdquo Journal of Difference Equations and Applications vol11 no 8 pp 735ndash758 2005

[17] J L Lions E Magenes and P Kenneth Non-HomogeneousBoundary Value Problems and Applications vol 1 SpringerBerlin Germany 1972

[18] R Uddin ldquoComparison of the nodal integral method andnonstandard finite-difference schemes for the Fisher equationrdquoSIAM Journal on Scientific Computing vol 22 no 6 pp 1926ndash1942 2000

[19] J Oh and D A French ldquoError analysis of a specialized numer-ical method for mathematical models from neurosciencerdquoApplied Mathematics and Computation vol 172 no 1 pp 491ndash507 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of